since differential-geometrically, the product 1+[y′⁢(x)]2⁢d⁢x is the arc-element. We rewrite (1) as

A=s⋅2⁢π⋅1s⁢∫sy⁢𝑑s.

Here, the last factor is the ordinate of the centroid of the rotating arc, whence we have the result

A=s⋅2⁢π⁢R

which states the first Pappus’s centroid theorem.

II. For deriving the second Pappus’s centroid theorem, we suppose that the region defined by

a≤x≤b,0≤y1⁢(x)≤y≤y2⁢(x),

having the area A and the centroid with the ordinate R, rotates about the x-axis and forms the solid of revolution with the volume V. The centroid of the area-element between the arcs y=y1⁢(x) and y=y2⁢(x) is [y2⁢(x)+y1⁢(x)]/2 when the abscissa is x; the area of this element with the width d⁢x is [y2⁢(x)-y1⁢(x)]⁢d⁢x. Thus we get the equation